Whakaoti i te {l}{-x+6y=20}{-x+3y=8}

$$\left. \begin{array} { l } { - x + 6 y = 20 } \\ { - x + 3 y = 8 } \end{array} \right.$$

Whakaoti mō x, y

$$x=4$$

$$y=4$$

Ngā Upane e Whakamahi ana i te Whakakapinga

Hei whakaoti i ētahi whārite takirua mā te whakakapinga, me whakaoti tētahi whārite i te tuatahi mō tētahi o ngā taurangi. Ka whakakapi i te otinga mō taua taurangi ki tērā o ngā whārite.

$$-x+6y=20,-x+3y=8$$

Kōwhiria tētahi o ngā whārite ka whakaotia mō te x mā te wehe i te x i te taha mauī o te tohu ōrite.

$$-x+6y=20$$

Me tango 6y mai i ngā taha e rua o te whārite.

$$-x=-6y+20$$

Whakawehea ngā taha e rua ki te -1.

$$x=-\left(-6y+20\right)$$

Whakareatia -1 ki te -6y+20.

$$x=6y-20$$

Whakakapia te 6y-20 mō te x ki tērā atu whārite, -x+3y=8.

$$-\left(6y-20\right)+3y=8$$

Whakareatia -1 ki te 6y-20.

$$-6y+20+3y=8$$

Tāpiri -6y ki te 3y.

$$-3y+20=8$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$-3y=-12$$

Whakawehea ngā taha e rua ki te -3.

$$y=4$$

Whakaurua te 4 mō y ki x=6y-20. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$x=6\times 4-20$$

Whakareatia 6 ki te 4.

$$x=24-20$$

Tāpiri -20 ki te 24.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=4$$

Ngā Upane e Whakamahi ana i Ngā Poukapa

Tuhia ngā whārite ki te tānga ngahuru ka whakamahi i ngā poukapa hei whakaoti i te pūnaha o ngā whārite.

$$-x+6y=20,-x+3y=8$$

Tuhia ngā whārite ki te tikanga tātai poukapa.

$$\left(\begin{matrix}-1&6\\-1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20\\8\end{matrix}\right)$$

Whakarea mauī i te whārite ki te poukapa kōaro o \left(\begin{matrix}-1&6\\-1&3\end{matrix}\right).

$$inverse(\left(\begin{matrix}-1&6\\-1&3\end{matrix}\right))\left(\begin{matrix}-1&6\\-1&3\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&6\\-1&3\end{matrix}\right))\left(\begin{matrix}20\\8\end{matrix}\right)$$

Ko te hua o tētahi poukapa me te kōaro ko te poukapa tuakiri.

$$\left(\begin{matrix}1&0\\0&1\end{matrix}\right)\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&6\\-1&3\end{matrix}\right))\left(\begin{matrix}20\\8\end{matrix}\right)$$

Whakareatia ngā poukapa kei te taha mauī o te tohu ōrite.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=inverse(\left(\begin{matrix}-1&6\\-1&3\end{matrix}\right))\left(\begin{matrix}20\\8\end{matrix}\right)$$

Mō te poukapa 2\times 2 \left(\begin{matrix}a&b\\c&d\end{matrix}\right), ko te \left(\begin{matrix}\frac{d}{ad-bc}&\frac{-b}{ad-bc}\\\frac{-c}{ad-bc}&\frac{a}{ad-bc}\end{matrix}\right) te poukapa kōaro, nō reira ka taea te tuhi anō te whārite poukapa hei rapanga whakarea poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}\frac{3}{-3-6\left(-1\right)}&-\frac{6}{-3-6\left(-1\right)}\\-\frac{-1}{-3-6\left(-1\right)}&-\frac{1}{-3-6\left(-1\right)}\end{matrix}\right)\left(\begin{matrix}20\\8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}1&-2\\\frac{1}{3}&-\frac{1}{3}\end{matrix}\right)\left(\begin{matrix}20\\8\end{matrix}\right)$$

Whakareatia ngā poukapa.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}20-2\times 8\\\frac{1}{3}\times 20-\frac{1}{3}\times 8\end{matrix}\right)$$

Mahia ngā tātaitanga.

$$\left(\begin{matrix}x\\y\end{matrix}\right)=\left(\begin{matrix}4\\4\end{matrix}\right)$$

Tangohia ngā huānga poukapa x me y.

$$x=4,y=4$$

Ngā Upane e Whakamahi i te Tangohanga

Hei whakaoti mā te tangohanga, ko ngā tau whakarea o tētahi o ngā taurangi me mātua ōrite i ngā whārite e rua kia whakakorehia ai te taurangi ina tangohia tētahi whārite mai i tētahi atu.

$$-x+6y=20,-x+3y=8$$

Me tango -x+3y=8 mai i -x+6y=20 mā te tango i ngā kīanga tau ōrite i ia taha o te tohu ōrite.

$$-x+x+6y-3y=20-8$$

Tāpiri -x ki te x. Ka whakakore atu ngā kupu -x me x, ka toe he whārite me tētahi taurangi kotahi ka taea te whakaoti.

$$6y-3y=20-8$$

Tāpiri 6y ki te -3y.

$$3y=20-8$$

Tāpiri 20 ki te -8.

$$3y=12$$

Whakawehea ngā taha e rua ki te 3.

$$y=4$$

Whakaurua te 4 mō y ki -x+3y=8. I te mea kotahi anake te taurangi kei te whārite i puta, ka taea e koe te whakaoti mō x hāngai tonu.

$$-x+3\times 4=8$$

Whakareatia 3 ki te 4.

$$-x+12=8$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$-x=-4$$

Whakawehea ngā taha e rua ki te -1.

$$x=4$$

Kua oti te pūnaha te whakatau.

$$x=4,y=4$$